Radiomic Feature Families

• Radiomic feature families are clearly defined groups of quantitative image features that capture tissue morphology, intensity, and spatial complexity.
• They incorporate standardized methods like IBSI definitions, convolution filtering, and local texture metrics to ensure reproducibility across studies.
• Standardized protocols and rigorous feature extraction enable linking computed features to clinical outcomes, enhancing predictive modeling in oncology.

Radiomic feature families comprise systematically defined groups of quantitative image features extracted from medical imaging data that capture distinct aspects of tissue morphology, intensity, and spatial complexity. These families enable reproducible, high-dimensional characterizations of regions of interest (ROIs) such as tumors. The major standards for radiomic features, particularly the Image Biomarker Standardisation Initiative (IBSI), designate formal classes and computational definitions, providing the foundation for interpretability, benchmarking, and robust radiomics-based modeling. Distinct feature families are further linked to computational complexity, semantic interpretability, and predictive performance in oncology and other clinical domains (Lei et al., 2020, Loutfi et al., 5 Jul 2024, Salmanpour et al., 14 Dec 2024).

1. Classification of Radiomic Feature Families

IBSI specifies eleven feature classes, which aggregate into major families differing in both image information content and computational procedure. Extensions in recent literature introduce further subdivisions based on filtering or local feature computation. The five-level taxonomy synthesizing IBSI and recent complexity-based approaches is as follows (Lei et al., 2020, Loutfi et al., 5 Jul 2024, Salmanpour et al., 14 Dec 2024):

Feature Family	Core Classes (IBSI)	Canonical Examples
Morphology (Shape)	3D Shape	Volume, surface area, compactness, sphericity, axis lengths
Intensity (First-order)	Local intensity, statistics, histogram, IVH	Mean, variance, percentiles, entropy, energy
Texture	GLCM, GLRLM, GLSZM, GLDZM, NGTDM, NGLDM/GLDM	Contrast, homogeneity, run entropy, zone emphasis
Linear-filtered	(Not isolated in IBSI; see convolution-filter extensions)	Wavelet, LoG, Gabor/Laws-filter responses
Nonlinear/local-filtered	(Windowed texture metrics, local GLCM/GLRLM)	Sliding window local contrast/entropy maps

Morphology and intensity families are invariant to filtering and spatial context; texture incorporates spatial arrangements of intensities via higher-order matrices. Filter-based families involve pre-processing images before standard feature computation, while nonlinear/local-filtered families compute features over local sub-volumes.

2. Mathematical Foundations and Formal Definitions

Radiomic families are characterized by precise mathematical definitions, typically expressed in terms of voxel intensities $x_i$, ROI mask, and (for texture) discretized levels and spatial relationships. Selected canonical definitions exemplify each family (Lei et al., 2020, Salmanpour et al., 14 Dec 2024, Loutfi et al., 5 Jul 2024):

Morphology (Shape):

• Volume: $V = \sum_{i \in \mathrm{ROI}} v_i$
• Surface area: $S = \sum_{t=1}^{N_{\text{tri}}}A_t$
• Sphericity: $\Psi = \frac{\pi^{1/3}(6V)^{2/3}}{S}$
• Axis lengths from inertia eigenvalues: $\operatorname{Major} = 2\sqrt{\lambda_1}$

Intensity (First-order):

• Mean: $\mu = \frac{1}{N}\sum_{i=1}^N x_i$
• Variance: $\sigma^2 = \frac{1}{N}\sum_{i=1}^N (x_i-\mu)^2$
• Skewness: $\frac{1}{N}\sum (\frac{x_i-\mu}{\sigma})^3$
• Entropy: $-\sum_{k} p_k \log p_k$
• Energy: $\sum_{i=1}^N x_i^2$

Texture (GLCM example):

Let $P(i,j)$ be the normalized gray-level co-occurrence for discrete values $i,j$.

• Contrast: $\sum_{i,j}(i-j)^2P(i,j)$
• Homogeneity (IDM): $\sum_{i,j} \frac{P(i,j)}{1+(i-j)^2}$
• Correlation: $$\frac{\sum_{i,j}(i-\mu_i)(j-\mu_j)P(i,j)}{\sigma_i\sigma_j}$$
• Joint entropy: $-\sum_{i,j}P(i,j)\log P(i,j)$

Linear-filtered and nonlinear-filtered features involve convolution with kernels $g$, e.g. LoG, Gabor, Haar/Daubechies wavelets, or locally windowed GLCM:

• Filtered image: $$h[\mathbf{k}_0]=(g*f)[\mathbf{k}_0]=\sum_{\mathbf{k}}g[\mathbf{k}]f[\mathbf{k}_0-\mathbf{k}]$$
• Windowed GLCM feature at voxel: $C(x,y,z) = \sum_{i,j}(i-j)^2\,P_{ij}^{(x,y,z)}$

Statistical, histogram, and texture family features are contingent upon discretization of intensity (fixed-bin or fixed-width). Discretization impacts texture features' sensitivity and reproducibility (Lei et al., 2020).

3. Standardization, Toolkits, and Reproducibility

IBSI establishes 173 standardized features across 11 classes, with rigorous mathematical definitions, digital phantoms, and reference values for benchmarking (Lei et al., 2020). Multiple software toolkits (PyRadiomics, MITK, LIFEx, SERA, CaPTk, etc.) implement subsets of these features, exhibiting high agreement for most intensity and texture features (relative differences $\mathrm{RD}\lesssim 1\%$), but poor reproducibility for morphologic/shape metrics (frequently $>5\%$ discrepancy due to mesh algorithm variance).

Two core "popularity" metrics quantify toolkit support:

• $P_1$ (average support fraction): Morphology $\approx0.16$, Intensity $\approx0.80$, Texture $\approx0.87$.
• $P_2$ (fraction implemented by at least 5/6 toolkits): Morphology $\approx0.03$, Intensity $\approx0.14$, Texture $\approx0.22$.

Morphology features are the least consistently implemented and most sensitive to software differences, whereas first-order and texture features achieve near-consensus when IBSI standards are followed.

A major source of feature disagreement is gray-level discretization: toolkits using distinct bin offsets (e.g., minimum = 0 vs 1) systematically shift first-order statistics and alter certain GLCM features (joint entropy, inverse difference moment). Consistent discretization is essential for cross-tool reproducibility (Lei et al., 2020).

4. Extended Feature Families: Filtering and Local Texture Maps

Recent taxonomies extend standard families by pre-filtering the image (linear filters) or by defining features on localized subvolumes (nonlinear/local-filtered features) (Depeursinge et al., 2020, Loutfi et al., 5 Jul 2024):

• Linear-filtered features apply convolutional kernels (mean, Gaussian, Laplacian of Gaussian, Gabor, Laws' kernels, (non)separable wavelets) to the image before feature extraction. Filtered response maps highlight structures at specific scales/orientations and aim to capture additional phenotype-relevant signal.
• Nonlinear/local-filtered features compute texture metrics (e.g., GLCM contrast, entropy) over sliding, windowed local patches, generating feature maps that reveal spatial heterogeneity.

The computational cost increases: morphology and first-order (O(N)), standardized texture (O(Nd+G²)), linear-filtered (O(Nk³) per filter), and nonlinear-filtered (O(Nw³G²))—with nonlinear filtering being the most intensive (Loutfi et al., 5 Jul 2024).

Features extracted from filter response maps are less reproducible than core IBSI features, demanding rigorous compliance testing with reference phantoms and full parameter reporting (filter type/ID, kernel size/scale, padding, voxel spacing, etc.) (Depeursinge et al., 2020).

5. Links to Biological Interpretation and Clinical Semantics

Feature families exhibit specific biological and radiological correlates, facilitating interpretability and clinical integration. For example, in the PM1.0 dictionary for prostate MRI (Salmanpour et al., 14 Dec 2024):

• First-order features (variance, entropy, percentiles) index tissue heterogeneity and intensity—linking to PI-RADS visual categories (hypo/hyperintensity, homogeneity) and cancer risk.
• Shape metrics (surface-to-volume ratio, sphericity, axis lengths) encode lesion geometry, relating directly to PI-RADS shape and size descriptors.
• Texture features (GLCM, GLRLM, GLSZM, NGTDM, GLDM) represent local spatial complexity: e.g., higher GLCM difference entropy or GLRLM run entropy associates with aggressive tumors or highly heterogeneous areas.

A compact, semantically annotated feature set—spanning minimal representatives from core families—has demonstrated high accuracy for prostate cancer risk stratification (Salmanpour et al., 14 Dec 2024).

6. Guidelines for Implementation, Validation, and Reporting

Rigorous reproducibility and interpretability depend on reference-conformant implementations, standardized reporting, and validation:

• Adopt IBSI terminology, mathematical formulas, and reference values.
• Always report software, version, programming language, meshing algorithm, discretization method, and feature aggregation strategy (Lei et al., 2020).
• Use digital phantoms/dataset with benchmark values as a compliance test after software installation/upgrades (Lei et al., 2020, Depeursinge et al., 2020).
• For filter-based features, resample to isotropic voxels, mirror-padding, and convert images to $\geq$32-bit float precision before filtering; report all filter parameters and boundary conditions (Depeursinge et al., 2020).
• Share feature definitions, source code, and extracted values publicly to facilitate meta-analysis and reproducibility (Lei et al., 2020).
• When selecting features for predictive modeling, select the "simplest" family (lowest computational complexity) that achieves maximal predictive power for the desired clinical endpoint—since added complexity rarely yields significant improvement for many endpoints (Loutfi et al., 5 Jul 2024).

Feature-family selection, standardization, and explicit methodological transparency are essential for generalizable, reproducible radiomic studies and trustworthy AI-based clinical decision support.